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A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities. (English) Zbl 1477.76055

Summary: In this work, we present a mass conservative numerical scheme for two-phase flow in porous media. The model for flow consists of two fully coupled, nonlinear equations: a degenerate parabolic equation and an elliptic one. The proposed numerical scheme is based on backward Euler for the temporal discretization and mixed finite element method for the spatial one. A priori stability and error estimates are presented to prove the convergence of the scheme. A monotone increasing, Hölder continuous saturation is considered. The convergence of the scheme is naturally dependant on the Hölder exponent. The nonlinear systems within each time step are solved by a robust linearization method, called the \(L\)-scheme. This iterative method does not involve any regularization step. The convergence of the \(L\)-scheme is rigorously proved under the assumption of a Lipschitz continuous saturation. For the Hölder continuous case, a numerical convergence is established. Numerical results (two-dimensional and three-dimensional) are presented to sustain the theoretical findings.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
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